# Exotic Mazur manifolds and knot trace invariants

## TL;DR

This paper constructs the first examples of Mazur 4-manifolds that are homeomorphic but not diffeomorphic, using knot Floer homology invariants to distinguish their smooth structures and exploring implications for 3-manifold surgeries.

## Contribution

It introduces the first pairs of exotic Mazur manifolds and demonstrates that the knot Floer invariant ν detects their smooth structure, unlike τ and ε.

## Key findings

- Existence of homeomorphic but not diffeomorphic Mazur manifolds.
- ν invariant distinguishes smooth structures of these manifolds.
- Construction of 3-spheres with two distinct S^1×S^2 surgeries.

## Abstract

From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant $\nu$ is an invariant of the smooth 4-manifold associated to a knot in the 3-sphere by attaching an n-framed 2-handle to the 4-ball along the knot. In contrast, we also show (modulo forthcoming work of Ozsv\'ath and Szab\'o) that the concordance invariants $\tau$ and $\epsilon$ are not invariants of such 4-manifolds. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct $S^1 \times S^2$ surgeries, resolving a question from Problem 1.16 in Kirby's list.